A127936 Numbers k such that 1 + Sum_{i=1..k} 2^(2*i-1) is prime.
1, 2, 3, 5, 6, 8, 9, 11, 15, 21, 30, 39, 50, 63, 83, 95, 99, 156, 173, 350, 854, 1308, 1769, 2903, 5250, 5345, 5639, 6195, 7239, 21368, 41669, 47684, 58619, 63515, 69468, 70539, 133508, 134993, 187160, 493095
Offset: 1
Examples
a(1)=1 because 1 + 2 = 3 is prime; a(2)=2 because 1 + 2 + 2^3 = 11 is prime; a(3)=3 because 1 + 2 + 2^3 + 2^5 = 43 is prime; a(4)=5 because 1 + 2 + 2^3 + 2^5 + 2^7 + 2^9 = 683 is prime; ...
Crossrefs
Programs
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Haskell
import Data.List (findIndices) a127936 n = a127936_list !! (n-1) a127936_list = findIndices ((== 1) . a010051'') a007583_list -- Reinhard Zumkeller, Mar 24 2013
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Mathematica
a = {}; Do[If[PrimeQ[1 + Sum[2^(2n - 1), {n, 1, x}]], AppendTo[a, x]], {x, 1, 1000}]; a b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, (1/2)(DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1])], {x, 1, Length[a]}]; d Position[Accumulate[2^(2*Range[1000]-1)],?(PrimeQ[#+1]&)]//Flatten (* The program generates the first 21 terms of the sequence. To generate more, increase the Range constant. *) (* _Harvey P. Dale, Mar 23 2022 *) -
PARI
for(n=1,999, ispseudoprime(2^(2*n+1)\3+1) & print1(n",")) \\ M. F. Hasler, Aug 29 2008
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Python
from sympy import isprime A127936 = [i for i in range(1,10**3) if isprime(int('01'*i+'1',2))] # Chai Wah Wu, Sep 05 2014
Formula
a(n) = floor(A000978(n)/2) = ceiling(log(4)(A000979(n))); A000978(n) = 2 a(n) + 1; A000979(n) = (2*4^a(n)+1)/3. - M. F. Hasler, Aug 29 2008
Extensions
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 11 2007
2 more terms from Stefan Steinerberger, Nov 24 2007
6 more terms from Dmitry Kamenetsky, Jul 12 2008
a(30)-a(40) calculated from A000978 by M. F. Hasler, Aug 29 2008
Comments